Efficient Algorithms For Solving Dynamic Responses Of Periodic Structures And Periodic Structures With Defects

A periodic structure is well known as comprising a number of identical unit cells joined together repetitively in an identical manner to form the entire structure.Because the periodic structure has the special structural properties,the physical characteristics and many advantages,such as low density,large modulus,high modulus,high specific strength,ease of manufacture and assembly.Therefore,the periodic structure is used in many fields such as engineering,aerospace,and new materials.With the development of science and technology,the scale of the periodic structures in the field of engineering and aerospace has become larger,and its structural form has become more and more complex and hence it it more difficult to analyze the periodic structure with high efficiency.Therefore,studying the periodic structure is very significant.In the time domain analysis,computing the dynamic responses of the periodic structures is always one of the issues of widespread concern.Now,there are many numerical algorithms for computing the structural dynamic responses,for example,the Newmark method,Runge-Kutta method,the central difference method,Generalized a method and Wilson-? method.For these numerical algorithms,the basic idea is that the dynamic equations corresponding to the entire structure is converted into a system of linear algebraic equations.Generally,the periodic structures contains a large number of unit cells.When the entire structure is discretized based on the finite element method in the spatial domain,the number of the DOFs corresponding to entire periodic structures is numerous,and the scale of the system of linear algebraic equations corresponding to the entire periodic structure is very large.Thus,it is very time-consuming for solving the system of linear algebraic equations corresponding to the entire periodic structure.Therefore,efficiently solving the system of linear algebraic equations corresponding to the entire periodic structures is the key issue for computing the dynamic responses of the periodic structures.In this PHD thesis,based on the condensation technology,Woodbury formula,the dynamic periodic structure properties and the group theory,several efficient numerical algorithms are presented for computing the dynamic responses of periodic structures and periodic structures with defects.The main research work includes the following three parts:(1)Based on the condensation technology,the Woodbury formula and the group theory,an efficient numerical algorithm for computing the dynamic responses of periodic structures is proposed.Based on the periodic properties of the structure and the condensation technology,the scale of the system of linear algebraic equations corresponding to the entire periodic structures is reduced.Thus,the efficiency of solving the system of linear algebraic equations is improved.By using the Woodbury formula,the solution of the system of linear algebraic equations corresponding to the periodic structure is obtained by solving a system of linear algebraic equations whose coefficient matrix corresponding to a cyclic periodic structures.Based on the group theory,the coefficient matrix of the system of linear algebraic equations corresponding to the cyclic periodic structures is transformed into a block-diagonal matrix,and hence the system of linear algebraic equations can be efficiently solved.The numerical examples show that:For the periodic structure with approximately 3.87 million DOFs,the efficiency of the proposed method is 5 times better than that of the direct method based on the Newmark method and 12 times better than that of the PCG method based on the Newmark method.(2)Based on the condensation technology,the dynamic periodic structure properties and the group theory,a more efficient numerical algorithm for computing the dynamic responses of periodic structures is proposed.Based on the properties of the system of linear algebraic equations that corresponds to the periodic structures,it is strictly proved in mathematics that,within a time step,the external force applied on any chosen unit cell can only influence a limited number of adjacent unit cells.And acoording to the process above,the number of the affected unit cells is determined.By using this conclusion,the dynamic responses of the entire periodic structures can be converted into the response analysis of some small-scale substructures.Furthermore,the dynamic responses of the small-scale substructures are converted into the response analysis of the cyclic periodic structures.Based on the group theory,the efficiency of computing the system of linear algebraic equations corresponding to the cyclic periodic structures can be further improved.The numerical examples show that:For the periodic structure with approximately 5.06 million DOFs,the efficiency of the proposed method is 11 times better than that of the direct method based on the Newmark method and 26 times better than that of the PCG method based on the Newmark method,and compare with the direct method and the PCG method based on the Newmark method,and the memory usage of the proposed method is very small.(3)Based on the condensation technology,the dynamic periodic structure properties and the group theory,an efficient numerical algorithm for computing the dynamic responses of periodic structures with defects is proposed.By using condensation technology,the system of linear equations corresponding to the entire periodic structures with defects can be obtained by the condensation of the system of linear equations that corresponds to one indentical unit cell and defective unit cells.Thus,the efficiency of computing the system of linear algebraic equation corresponding to the entire structure with defects is improved.By using the conclusion that within a time step,the external force applied on any chosen unit cell can only influence a limited number of adjacent unit cells,the dynamic responses of the periodic structures with defects are converted into the response analysis of a small-scale substructure with defects and a perfect periodic structure.Because the scale of the substructure with defects is very small,the dynamic responses of the small-scale subtructure with defects can be efficiently solved.And the dynamic responses of the perfect periodic structure can be efficiently computed by using the numerical method proposed in this PHD paper.The numerical examples show that:For the periodic structure with defects and approximately 5.34 million DOFs,the efficiency of the proposed method is more than 10 times better than that of the direct method based on the Newmark method and more than 40 times better than that of the PCG method based on the Newmark method,and the numerical method proposed in this paper saves memory.